If $\mathscr{E}$ is a vector bundle of rank $k$ and $\mathcal{L}$ is a line bundle, Then: $$c_{k}(\mathscr{E} \otimes \mathcal{L}) = \displaystyle \sum_{i = 0}^{k}\binom{r-l}{k-l}c_{1}(\mathcal{l})^{k-l}c_{l}(\mathscr{E})$$ Is there a similar result for the tensor product of coherent sheaves?
For example, if $r_{1} = \mbox{rank}(E)$ and $r_{2} = \mbox{rank}(F)$, then: $$c_{1}(E \otimes F) = r_{2}c_{1}(E) + r_{1}c_{1}(F)$$ $$c_{2}(E \otimes F)?$$ $$\cdots$$ $$c_{k}(E \otimes F)?$$ where $E$ and $F$ are coherent sheaves (we can assume them in a smooth projective variety of dimension $n$).
Thank you very much.